David Cameron has delivered a blunt warning to the Syrian president, Bashar al-Assad, that he must leave power immediately or watch his country descend into civil war.

In some of the first remarks by a western leader after the assassination of members of the Assad inner circle in Damascus, the prime minister said a lack of change at the top would lead to revolution from the bottom in Syria.

The prime minister, who was speaking at a press conference in Kabul with the Afghan president, Hamid Karzai, also urged Russia to "step up to the plate" and stop vetoing UN sanctions against Assad.

Asked what his message was to Assad after the escalation of violence in Syria, the prime minister said: "I would have a very clear message for President Assad: it is time for him to go. It is time for transition in this regime. Clearly Britain does not support violence on either side. But if there isn't transition, it is quite clear there is going to be civil war. That is the clear fact we can all see on the ground.

"The regime has done some truly dreadful things to its own people. I don't think any regime that carries out acts as they have against their own citizens, and continues to do so, should survive. That regime should go."

Cameron highlighted British and EU impatience with Vladimir Putin, who has vetoed a series of proposed UN security council resolutions that would have imposed tough sanctions on Syria. Russia has been joined by China, one of five permanent members of the UN security council which have the power of veto. Britain, the US and France also have vetoes.

Cameron said: "The message to President Putin and to all those on the UN security council is: it is time for the UN security council to pass clear and tough messages about sanctions – I believe under chapter seven of the UN – and to be unambiguous in this.

"Obviously we are a UN security council with permanent members and permanent members that have vetoes. We can't pass these things without everybody stepping up to the plate and taking the right action. But I would appeal to those who in the past have held out against tough action against Syria: what more evidence do we need about a regime that had brutalised its people?"

The prime minister said there was a clear alternative facing Syria. "The alternative to political transition at the top of Syria is revolution from the bottom in Syria. It is in everybody's interests – the Syrian people, the region, the wider world, the fight against terrorism – that transition takes place quickly. The sooner that happens, the sooner the people of Syria can be freed from the tyranny under which they are currently suffering."

Diplomats are likely to note that the prime minister said he would like a new UN security council resolution to be agreed under chapter seven of the UN charter, which authorises the use of force. British sources said Cameron believes this would provide crucial legal authority for non-military intervention in Syria. But invoking chapter seven could provide legal cover for military intervention at a later stage, though this would depend on the wording of the security council resolution. The sources pointed out that chapter seven is mentioned in the draft resolution.

Russia has been wary of supporting a new UN resolution on Syria partly because it believes that Britain and France stretched the resolution agreed on Libya. This authorised military action to protect civilians from aggressive attacks by the regime in Tripoli. Moscow believes that London and Paris bent the law to use the security council to overthrow Muammar Gaddafi.